This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian systems, which includes nonlinear wave equations. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler-Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler-Bernoulli beam model, the sine-Gordon (nonlinear) wave equation, and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. The learned Lagrangian ROMs generalize well, as they can accurately predict the physical solutions both far outside the training time interval, as well as for unseen initial conditions.

翻译：本文提出了一种无侵入的保物理方法，用于学习拉格朗日系统（包括非线性波动方程）的降阶模型（ROM）。现有的侵入式投影降阶方法通过将全阶模型（FOM）的欧拉-拉格朗日方程投影到线性子空间上，构造保结构的拉格朗日ROM。这种伽辽金投影步骤要求完全了解FOM中的拉格朗日算子，并能全面操作计算机代码。相比之下，本文提出的拉格朗日算子推断方法将力学原理嵌入算子推断框架中，开发了一种数据驱动的模型降阶方法，可保留底层拉格朗日结构。该方法利用对控制方程（而非其离散化形式）的认知，定义拉格朗日ROM的形式与参数化，进而通过投影快照数据学习该ROM。该方法无需访问FOM算子或计算机代码。数值结果展示了拉格朗日算子推断在欧拉-伯努利梁模型、正弦-戈登（非线性）波动方程以及具有779,232自由度的软体机器人鱼尾大规模离散化系统上的应用。所学的拉格朗日ROM具有良好的泛化能力，既能准确预测训练时间区间之外的物理解，也能对未见过的初始条件进行准确预测。